library(tidyr)
library(ggplot2)
library(corrplot)
## corrplot 0.84 loaded
library(GGally)

2. Loading the Boston data and exploring the data

library(MASS)
data("Boston")
str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00

According to the structure of the data, Boston dataset, which is already loaded in R, has 506 observattions (rows) and 14 variables (columns). The data are mainly gathered to understand the effect of housing values in suburbs of Boston on different vairbles such as crime rate.

3. Graphical overview of the data and summary of variables

pairs(Boston)

The plot above shows the correlations of all the variables in the dataset. But let’s look at a more advanced plot to see the distribution of the data as well as correlation of the variables.

p <- ggpairs(Boston, mapping = aes(alpha = 0.3), lower = list(combo = wrap("facethist", bins = 20)))
p

The above plot demonstrates the distribution of the data as well as the correlations of the variables. As an example, data in the variable “rm”, average number of room per dwelling, have normal distribution, and there is a positive, and rather strong, correlation between “zn” (proportion of residential land zoned for lots over 25,000 sq.ft.), and “dis” (weighted mean of distances to five Boston employment centres).

But we can look at the corraltions with more r funcations. Let’s explore cor_matrix function, which provides handy, easy-to-interpret-correlation matrix.

cor_matrix<-cor(Boston) %>% round(digit=2)

cor_matrix %>% round(cor_matrix)
##         crim  zn indus chas nox  rm age dis rad tax ptratio black lstat
## crim     1.0 0.0   0.0    0 0.0 0.0 0.0 0.0 0.6 0.6       0     0   0.0
## zn       0.0 1.0   0.0    0 0.0 0.0 0.0 0.7 0.0 0.0       0     0   0.0
## indus    0.0 0.0   1.0    0 0.8 0.0 0.6 0.0 0.6 0.7       0     0   0.6
## chas     0.0 0.0   0.0    1 0.0 0.0 0.0 0.0 0.0 0.0       0     0   0.0
## nox      0.0 0.0   0.8    0 1.0 0.0 0.7 0.0 0.6 0.7       0     0   0.6
## rm       0.0 0.0   0.0    0 0.0 1.0 0.0 0.0 0.0 0.0       0     0   0.0
## age      0.0 0.0   0.6    0 0.7 0.0 1.0 0.0 0.0 0.5       0     0   0.6
## dis      0.0 0.7   0.0    0 0.0 0.0 0.0 1.0 0.0 0.0       0     0   0.0
## rad      0.6 0.0   0.6    0 0.6 0.0 0.0 0.0 1.0 0.9       0     0   0.0
## tax      0.6 0.0   0.7    0 0.7 0.0 0.5 0.0 0.9 1.0       0     0   0.5
## ptratio  0.0 0.0   0.0    0 0.0 0.0 0.0 0.0 0.0 0.0       1     0   0.0
## black    0.0 0.0   0.0    0 0.0 0.0 0.0 0.0 0.0 0.0       0     1   0.0
## lstat    0.0 0.0   0.6    0 0.6 0.0 0.6 0.0 0.0 0.5       0     0   1.0
## medv     0.0 0.0   0.0    0 0.0 0.7 0.0 0.0 0.0 0.0       0     0   0.0
##         medv
## crim     0.0
## zn       0.0
## indus    0.0
## chas     0.0
## nox      0.0
## rm       0.7
## age      0.0
## dis      0.0
## rad      0.0
## tax      0.0
## ptratio  0.0
## black    0.0
## lstat    0.0
## medv     1.0
corrplot(cor_matrix, method="circle", type = "upper", cl.pos = "b", tl.pos = "d", tl.cex = 0.6)

Let’s explore some examples. We can see that there is a strong and positive relationship between “rad” (index of accessibility to radial highways) and “tax” (full-value property-tax rate per $10,000). In addition, there is a negtive and strong relationship between “lstat” (lower status of the population (percent)) and “medv” (median value of owner-occupied homes in $1000s).

4. Standardizing the dataset

boston_scaled <- scale(Boston)
summary(boston_scaled)
##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv        
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865
class(boston_scaled)
## [1] "matrix"
boston_scaled <- as.data.frame(boston_scaled)

Creating a categorical variable of the crime rate in the Boston dataset. First, let’s create a quantile vector of crime rate.

bins <- quantile(boston_scaled$crim)
bins
##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610

Now, let’s create a categorical variable, and call it, ‘crime’:

crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, c(label = "low", "med_low", "med_high", "high"))
table(crime)
## crime
##      low  med_low med_high     high 
##      127      126      126      127

Now, let’s remove the original “crim”" from the dataset

boston_scaled <- dplyr::select(boston_scaled, -crim)

And add the new categorical value to scaled data

boston_scaled <- data.frame(boston_scaled, crime)

Dividing the dataset to train and test sets, so that 80% of the data belongs to the train set.

n <- nrow(boston_scaled)
ind <- sample(n,  size = n * 0.8)
train <- boston_scaled[ind,]
test <- boston_scaled[-ind,]
correct_classes <- test$crime
test <- dplyr::select(test, -crime)

5. Fitting the linear discriminant analysis (LDA) on the train dataset

Use the crime as a target variable and all the other variables as predictors.

lda.fit <- lda(crime ~ ., data = train)
lda.fit
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2425743 0.2475248 0.2549505 0.2549505 
## 
## Group means:
##                  zn      indus         chas        nox         rm
## low       0.8738901 -0.8974583 -0.071456607 -0.8852769  0.3688227
## med_low  -0.1127084 -0.2809154  0.003267949 -0.5622831 -0.1158019
## med_high -0.3773415  0.1921577  0.071689400  0.4113317  0.1105528
## high     -0.4872402  1.0170891 -0.119431971  1.0399658 -0.3863412
##                 age        dis        rad        tax      ptratio
## low      -0.8893100  0.8172347 -0.6830354 -0.7424175 -0.422512835
## med_low  -0.3018559  0.3494348 -0.5477507 -0.4470063 -0.003941714
## med_high  0.4462700 -0.3657065 -0.4076377 -0.3010156 -0.331495417
## high      0.8126991 -0.8715363  1.6384176  1.5142626  0.781113578
##                black       lstat        medv
## low       0.38298362 -0.72641147  0.44899335
## med_low   0.31496261 -0.11165892 -0.02498681
## med_high  0.08519867 -0.04740194  0.17515111
## high     -0.81330145  0.89390654 -0.67684776
## 
## Coefficients of linear discriminants:
##                 LD1          LD2         LD3
## zn       0.13497819  0.636804742 -0.99191549
## indus    0.01874516 -0.397110369  0.17434722
## chas    -0.01504716  0.040628075  0.16892374
## nox      0.43756953 -0.792413542 -1.19409903
## rm       0.03142982 -0.069805285 -0.08944692
## age      0.27572113 -0.494435411 -0.05866500
## dis     -0.07725345 -0.395629469  0.26997267
## rad      3.17470222  0.777476587 -0.40551909
## tax     -0.03982591  0.219629786  0.78239173
## ptratio  0.13510805 -0.006898625 -0.10993219
## black   -0.11793010  0.018541782  0.08538876
## lstat    0.15935078 -0.071577781  0.53951334
## medv     0.05595666 -0.381757587 -0.04065595
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9475 0.0400 0.0124

Drawing the LDA (bi)plot. But first let’s create a numeric vector of the train sets crime classes.

classes <- as.numeric(train$crime)

Now we’re ready to draw the LDA plot.

lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 3)

6. Predicting the classes with the LDA model

In part 4, categorical crime variable has already been removed from the test dataset, and we have now the test data and correct class labels. Now, we predict the classes with the LDA model on the test data.

lda.pred <- predict(lda.fit, newdata = test)

Cross tabulating the results by creating a table of the correct classes and the predicted ones.

table(correct = correct_classes, predicted = lda.pred$class)
##           predicted
## correct    low med_low med_high high
##   low       20       7        2    0
##   med_low    6      18        2    0
##   med_high   0       9       13    1
##   high       0       0        0   24

The table above reveals that in most cases the crime rates have been predicted correctly, but there are some inconsistencies between the predicted results and correct ones. It seems that the class, high (crime rates), is the most precisely predicted one.

7. Calculating the distances between the observations

First, reloading and standardizing the dataset.

data('Boston')
boston_scaled <- scale(Boston)
boston_scaled <- as.data.frame(boston_scaled)

Now, let’s use dist() function to calculate the distances between observation using the most common distance measure, which is Euclidean distance.

dist_eu <- dist(Boston)
summary (dist_eu)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.119  85.624 170.539 226.315 371.950 626.047

Now, let’s use manhattan distance matrix, another distance measure.

dist_man <- dist(Boston, method = "manhattan")
summary(dist_man)
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
##    2.016  149.145  279.505  342.899  509.707 1198.265

Now, we investigate the optimal number of clusters, using K-means clustering. First, we run K-means algorithm on the data.

km <-kmeans(Boston, centers = 4)
pairs(Boston, col = km$cluster)

Now, we investigate the optimal number of clusters using K-means clustering, with 10 clusters. One way to specify the number of clusters is to look at how the total of within cluster sum of squares (WCSS) behaves when the number of cluster changes. So, let’s look at the behavior of WCSS and plot it.

set.seed(123)
k_max <- 10
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})
qplot(x = 1:k_max, y = twcss, geom = 'line')

By visualizing the total WCSS as a graph, we can see that two clusters would seem optimal. Becasue, the optimal number of clusters is when the value of total WCSS changes radically.
So, let’s run k-means again with two clusters and then visualize the results.

km <-kmeans(Boston, centers = 2)
pairs(Boston, col = km$cluster)

Creating a 3D plot

model_predictors <- dplyr::select(train, -crime)

check the dimensions

dim(model_predictors)
## [1] 404  13
dim(lda.fit$scaling)
## [1] 13  3

matrix multiplication

matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
library(plotly)
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:MASS':
## 
##     select
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, col = crime, type= 'scatter3d', mode='markers')
## Warning: 'scatter3d' objects don't have these attributes: 'col'
## Valid attributes include:
## 'type', 'visible', 'showlegend', 'legendgroup', 'opacity', 'name', 'uid', 'ids', 'customdata', 'hoverinfo', 'hoverlabel', 'stream', 'x', 'y', 'z', 'text', 'hovertext', 'mode', 'surfaceaxis', 'surfacecolor', 'projection', 'connectgaps', 'line', 'marker', 'textposition', 'textfont', 'error_x', 'error_y', 'error_z', 'scene', 'xcalendar', 'ycalendar', 'zcalendar', 'idssrc', 'customdatasrc', 'hoverinfosrc', 'xsrc', 'ysrc', 'zsrc', 'textsrc', 'hovertextsrc', 'textpositionsrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule'
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', col = km$cluster)
## Warning: 'scatter3d' objects don't have these attributes: 'col'
## Valid attributes include:
## 'type', 'visible', 'showlegend', 'legendgroup', 'opacity', 'name', 'uid', 'ids', 'customdata', 'hoverinfo', 'hoverlabel', 'stream', 'x', 'y', 'z', 'text', 'hovertext', 'mode', 'surfaceaxis', 'surfacecolor', 'projection', 'connectgaps', 'line', 'marker', 'textposition', 'textfont', 'error_x', 'error_y', 'error_z', 'scene', 'xcalendar', 'ycalendar', 'zcalendar', 'idssrc', 'customdatasrc', 'hoverinfosrc', 'xsrc', 'ysrc', 'zsrc', 'textsrc', 'hovertextsrc', 'textpositionsrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule'